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Properties

Label 2-946-11.3-c1-0-2
Degree $2$
Conductor $946$
Sign $-0.745 + 0.666i$
Analytic cond. $7.55384$
Root an. cond. $2.74842$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.972 + 2.99i)3-s + (0.309 − 0.951i)4-s + (1.26 + 0.917i)5-s + (−2.54 − 1.84i)6-s + (−0.230 + 0.708i)7-s + (0.309 + 0.951i)8-s + (−5.58 + 4.05i)9-s − 1.56·10-s + (−1.40 − 3.00i)11-s + 3.14·12-s + (−2.96 + 2.15i)13-s + (−0.230 − 0.708i)14-s + (−1.51 + 4.67i)15-s + (−0.809 − 0.587i)16-s + (−5.72 − 4.15i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.561 + 1.72i)3-s + (0.154 − 0.475i)4-s + (0.564 + 0.410i)5-s + (−1.03 − 0.755i)6-s + (−0.0869 + 0.267i)7-s + (0.109 + 0.336i)8-s + (−1.86 + 1.35i)9-s − 0.493·10-s + (−0.424 − 0.905i)11-s + 0.908·12-s + (−0.821 + 0.597i)13-s + (−0.0615 − 0.189i)14-s + (−0.392 + 1.20i)15-s + (−0.202 − 0.146i)16-s + (−1.38 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(7.55384\)
Root analytic conductor: \(2.74842\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{946} (861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 946,\ (\ :1/2),\ -0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318326 - 0.834454i\)
\(L(\frac12)\) \(\approx\) \(0.318326 - 0.834454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (1.40 + 3.00i)T \)
43 \( 1 + T \)
good3 \( 1 + (-0.972 - 2.99i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.26 - 0.917i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.230 - 0.708i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.96 - 2.15i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.72 + 4.15i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.247 + 0.760i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + (1.71 - 5.27i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.63 - 4.09i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.62 - 8.06i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.99 - 9.22i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (-1.56 - 4.81i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.20 + 3.05i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.21 - 12.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.93 + 4.30i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.13T + 67T^{2} \)
71 \( 1 + (-6.75 - 4.90i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.74 + 5.36i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.86 - 1.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.54 - 4.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 + (-10.1 + 7.34i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.42729312141839762425604830093, −9.494614761511264375648605153002, −9.101060369884872364818218204688, −8.473555152246294721844066433305, −7.25895478803187827411627394067, −6.24947677273255149273883321975, −5.16994047740382676869622337150, −4.60035434910474355222329225305, −3.14862133432826842093237739255, −2.41914880500254809290845004167, 0.42513063285805390015391064166, 2.03650821941686587054691138890, 2.19897255472904940903563334944, 3.78044616101594264882739369700, 5.35602889884164185165331949681, 6.39740107084820869731883893519, 7.38041018022996863494321440326, 7.63912860149102437213908954548, 8.775140142720341578747705452270, 9.249644986313540470242443612037

Graph of the $Z$-function along the critical line